# Rainbow AP(4) in an almost equinumerous coloring

 Importance: Medium ✭✭
 Author(s): Conlon, David
 Subject: Combinatorics
 Keywords: arithmetic progression rainbow
 Posted by: vjungic on: July 20th, 2007
Problem   Do 4-colorings of , for a large prime, always contain a rainbow if each of the color classes is of size of either or ?

It is known that there are equinumerous colorings of (i.e. colorings of for some such that each color occurs times) within which we cannot find rainbow arithmetic progressions of length . ([CJR])

## Bibliography

*[C] David Conlon, Rainbow solutions of linear equations over , Discrete Mathematics, 306 (2006) 2056 - 2063.

[CJR] David Conlon, Veselin Jungic, Rados Radoicic, On the existence of rainbow 4-term arithmetic progressions, Graphs and Combinatorics, 23 (2007), 249-254

* indicates original appearance(s) of problem.

### Tight hypergraphs

It deservs to be mentioned that in any -colouring of , the equation , have an heterochromatic (rainbow) solution; here, denotes the multiplicative group of the field .